cp's OEIS Frontend

A213896 Fixed points of a sequence h(n) defined by the minimum number of 7's in the relation n*[n,7,7,...,7,n] = [x,...,x] between simple continued fractions.

%I A213896 #22 Sep 16 2015 09:23:20
%S A213896 2,3,19,31,67,79,103,127,139,151,167,179,191,263,283,359,383,443,463,
%T A213896 479,491,503,571,631,691,787,827,883,919,1019,1087,1171,1291,1303,
%U A213896 1307,1327,1399,1423,1451,1487
%N A213896 Fixed points of a sequence h(n) defined by the minimum number of 7's in the relation n*[n,7,7,...,7,n] = [x,...,x] between simple continued fractions.
%C A213896 In a variant of A213891, multiply n by a number with simple continued fraction [n,7,7,..,7,n] and increase the number of 7's until the continued fraction of the product has the same first and last entry (called x in the NAME). Examples are
%C A213896 2 * [2, 7, 7, 2] = [4, 3, 1, 1, 3, 4],
%C A213896 3 * [3, 7, 7, 7, 3] = [9, 2, 2, 1, 1, 1, 2, 2, 9] ,
%C A213896 4 * [4, 7, 7, 7, 7, 7, 4] = [16, 1, 1, 3, 1, 1, 1, 6, 1, 1, 1, 3, 1, 1, 16],
%C A213896 5 * [5, 7, 7, 5] = [25, 1, 2, 2, 1, 25] ,
%C A213896 6 * [6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 6] = [36, 1, 5, 3, 1, 4, 10, 1, 2, 2, 4, 2, 2, 1, 10, 4, 1, 3, 5, 1, 36],
%C A213896 7 * [7, 7, 7] = [49, 1, 49] .
%C A213896 The number of 7's needed defines the sequence h(n) = 2, 3, 5, 2, 11, 1, 5, 11, 2,...  (n>=2).
%C A213896 The current sequence contains the fixed points of h, i. e., those n where h(n)=n.
%C A213896 We conjecture that this sequence contains prime numbers analogous to the sequence of prime numbers A000057, in the sense that, instead of referring to the Fibonacci sequences (sequences satisfying f(n)=f(n-1)+f(n-2) with arbitrary positive integer values for f(1) and f(2)) it refers to the sequences satisfying f(n)=7*f(n-1)+f(n-2), A054413, A015453, etc. This would mean that a prime is in the sequence A213896 if and only if it divides some term in each of the sequences satisfying f(n)=7*f(n-1)+f(n-2).
%C A213896 The above sequence h() is recorded as A262217. - _M. F. Hasler_, Sep 15 2015
%o A213896 (PARI)
%o A213896 {a(n) = local(t, m=1); if( n<2, 0, while( 1,
%o A213896    t = contfracpnqn( concat([n, vector(m,i,7), n]));
%o A213896    t = contfrac(n*t[1,1]/t[2,1]);
%o A213896    if(t[1]<n^2 || t[#t]<n^2, m++, break));
%o A213896 m)};
%o A213896 for(k=1,1500,if(k==a(k),print1(a(k),", ")));
%Y A213896 Cf. A000057, A213891 - A213895, A213897 - A213899, A261311; A213358.
%Y A213896 Cf. A213648, A262212 - A262220, A213900, A262211.
%K A213896 nonn
%O A213896 1,1
%A A213896 _Art DuPre_, Jun 23 2012